[FOM] Re: Constructive analysis

[Joseph Miller]

> Ayan Mahalanobis writes:
> 
> As I understand it (again correct me if I am wrong) BISH is more like 
> doing classical mathematics constructively and I sometime wonder why would 
> a constructivist be interested in that. The fundamental reason of doing 
> constructive mathematics is meaning as I understood it which is a product 
> of dissatisfaction from classical math. Then to embrace it as a guideline 
> is self-defeating to me.

Perhaps the constructivists should speak for themselves, but I would guess
that they do not feel that a result which was first proved using classical
techniques has been sullied by the association, but that it had not been
proved at all. 

The recovery of parts of classical mathematics (perhaps with slight
modifications) by constructivists seems analogous to giving Calculus
foundations independent of infinitesimals, which were once thought to be
outside of safe and rigorous mathematical reasoning.  In other words, the
theorems didn't change as much as the notion of what constituted a valid
proof. 

That said, I agree that Bishop is more interested in justifying old ideas
than was the Russian school.  Not only did he not work with any principals
inconsistent with classical mathematics, but he was careful to take
hypotheses that gave analogs to classical theorems where it was
reasonable.  For example, he restricted his attention to continuous
functions which are (constructively) uniformly continuous on finite closed
intervals.  This eliminates several of the more obnoxious pathologies
enjoyed by the Russian school, but doesn't seem to me to be a cop-out as
much as a correction of (what he might have seen as) classical
carelessness. 

Joseph Miller
Indiana University